where \(\eta \)  is a scalar convex entropy function and \(\zeta _{j}\)  are
associated entropy fluxes. This assumption is valid for many systems of
balance laws from continuum physics which typically have a natural companion
balance law related to the second law of thermodynamics, e.g.,
[1]}. Specifically, companion balance
law (REF ) provides an entropy balance constraint on
the solution of balance law (REF ). The vector of
entropy variables \(\mathbf {\beta }\)  is obtained by differentiating the entropy
function with respect to the state \(\mathbf {q}\) . That is, \(\mathbf {\beta } =\frac{\partial \eta }{\partial \mathbf {q}}\) , with component \(\alpha = 1, \dots ,N_{c}\)  given by
\(\beta _{\alpha }(\mathbf {q}, \mathbf {x})= \frac{\partial \eta (\mathbf {q}, \mathbf {x})}{\partial q_{\alpha }}.\) 
